Big Groups and Big Products

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SWBAT apply their understanding of 1 digit by 1 digit multiplication to multiply a 2 digit number by a 1 digit number involving larger groups and larger products.

Big Idea

Big groups and big products can be broken apart into smaller groups of tens and ones to make it easier to multiply and find the product

Lesson Introduction

10 minutes

My students have a strong foundation of creating equal groups of whole numbers when multiplying single digit factors and are ready to break apart 2 digit by 1 digit multiplication to understand larger groups. Critical Area #1 in the Common Core calls for students to develop a deep understanding of multiplication through activities and problems involving equal sized groups and arrays, which we will do. I am going to introduce how to multiply a 2 by 1 digit number and how we can connect what we are learning to our knowledge of a base 10 system and to creating equal groups when multiplying. At times my students do not line up their numbers correctly so I will refer to their hundreds, tens and ones places to reemphasize the importance of place value even when multiplying. I also always spend a minute or two activating what they know and what this connects to in order to build a stronger foundation.

Today we are going to get into something that real math experts do well. I think you guys are ready for it because I’ve seen how hard you’ve worked on multiplication so far. Who can remind me what I can use a hundreds, tens and ones place value chart for when I multiply?

Make sure students are able to explain how place value is used in all math and that the place changes the value of a number, or similar types of understanding about a base 10 system.

Even math experts must make sure that numbers are in their place! Now, when we think about multiplying, what are we really trying understand?

I stop my instruction for a few minutes now, because it is important that I find out what my students understand before proceeding. I am checking to see if they can identify that we are repeatedly adding a number, that we are creating equal groups of objects and that we can use skip counting strategies. I am also checking to ensure that students are relating place value to multiplication (a 9 in the tens place when we multiply is really a 90 multiplied by my single digit). The common core references the balance between procedure and understanding, and the understanding at this point is critical for the progression into division and fact families. Without this understanding, students will not be able to break apart equal groups or repeatedly subtract to find solutions when we move into division.

Wow, I knew that you guys were ready for this, and I was right. When we look at multiplying larger groups, or more things in each group, we can set it up just like we do when we add, subtract, or even when we multiply 1 digit numbers together. 

Guided Practice

10 minutes

I uncover a 2 digit by 1 digit multiplication problem on the board and ask students to think about how they can use what they know about 1 digit by 1 digit multiplication problems to help them solve this problem. We have done a lot of work in 1 digit multiplication to show that we are creating equal groups. Many students draw out their groups and put dots or other symbols to model what is in each group. I don’t move on to 2 by 1 digit multiplication until students have a firm foundation of creating groups, what is within each group and what total numbers of objects we have as a product. I ask them to turn and talk to a partner about how they might solve this problem, and then I bring them back together and ask for suggestions. Most of my students refer to creating groups, putting something in each group and finding the total number of objects. Some students may express that using repeated addition of 15, 3 times gives them 35. Mathematical Practice 8 highlights that students should look for repetition and look for more efficient ways to solve, so this is important to emphasize. Others may remember how to regroup and carry with the traditional algorithm. I note these responses on the chart and ask them what type of model helps them solve the problem, which I draw out next to their answer. I have a lot of ELL’s in my class so I always try to create or represent a problem with numbers and with a picture or model.

I’m hearing some great thinking going on, so I’m ready to show you one way I think about 2 digit by 1 digit multiplication. I remember hearing you tell me that multiplication was equal groups of things, but when I multiply these numbers my picture just gets too big and I lose count. I’m going to break my tens and ones places to make multiplying easier and to make my picture less confusing.

I walk students through multiplying my numbers in the ones place and then I ask them what I use that I already know about multiplying to help me find my solution. I draw both the 3 groups with 5 in each and then highlight that I can also skip count by my 5’s 3 times and record this on the chart paper. I move to multiplying my single digit by my number in the tens place.

Ok, now that I’ve recorded my product of my ones place, I’ve got to move next door to my tens place. When I multiply my single digit by the number in my tens place, let’s see if my equal groups and repeated addition also work here. I have a 3 and a 1. What do I know about multiplication that will help me solve this? Can I also use a picture model to represent the product?

It is important at this point that students understand the 1 represents a group of 10 and so I don’t have 3 of something, but 30. 

Independent Practice

30 minutes

I use a lot of number cards and dice to practice multiplication facts because it allows my students to manipulate something to make each problem, it feels like a game, and they have a random assortment of problems that come up. Because this is day 1 of 2 by 1 digit multiplication we are only practicing the multiplication procedure and representing the problem with models and repeated addition.  

Well, it looks like you’re on your way to becoming experts already! As you return to your tables to practice your new skills, make sure you are thinking about the ways that we use in 2 digit by 1 digit multiplication to break apart our tens and our ones place to make it easier to find our product. If you get stuck, we have our examples on the chart in case you need to see another way of thinking about your problem.

At this point I release students to tables where they have a pile of number cards with 2 digits, and a pile of number cards that have 1 digit face down on their tables.  Students pull cards to create problems, and record their work in their journals. The common core calls for students to understand that multiplication is equal groups of objects, so it is important that students can justify their answers using an array, model or picture. I walk around the room and lean in to students to ask questions about their work, which helps me understand where conceptual breakdowns are occurring.

Key questions: Why is ___ x ___ = ?, what does this model/array represent? Why did you choose to set up your problem in this way? Can you explain how you solved this problem to me?


5 minutes

Wow, I’m blown away by your work! Today we took what we know about multiplication and place value to multiply 2 digit by 1 digit numbers to make larger groups or objects easier to understand. Who can share something else they discovered?

I will also call on someone to share an example of these work, which I have attached images of in the Independent Practice section. I think it is important for students to share what they learned and for students to learn from one another. My students love to share and love to celebrate great work.

I think it is critical to provide a closure to your lesson. I highlight what we learned by restating what was covered and asking students to share what they discovered today in the lesson. This allows me to check for understanding, which will allow me to inform my instruction for tomorrow. At times I have students complete 1 problem on a note card and hand it in as a check for understanding on the way out the door. This allows me to look over it later and pull a small group for tomorrow based on who does not have an understanding of the repeated addition, equal groups or who needs additional work with place value. 

 View the example of the student explaining their work